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Credit goes to PhysicsOverflow user berceanu for the summary: This paper applies quite generally to noninteracting particles described by a Hamiltonian containing a translationally invariant (or periodic) part and a weak, slow varying additional potential. It is shown that this type of Hamiltonian corresponds to the momentum space dual of the Hamiltonian of a charged particle in an EM field. The paper then explores the role played by the topology of momentum space in such systems. In particular, for the concrete example of a particle (described by the Harper-Hofstadter model) which hops on a 2D lattice in the presence of an external harmonic potential, it is shown that the eigenspectrum can be physically interpreted as the energy levels of a particle in constant magnetic field moving on the surface of a torus (Landau levels on a torus). Finally, different possible setups are proposed for the experimental realisation, both in a cold atom and photonic context.

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The main subject of this paper is the pretty analogy between the k-space Berry's phase and x-space magnetic fields. This analogy appears whenever you work with Berry's phase for k-space in periodic crystals, for instance, you can see it explicitly in the strange classical equation of motion (natural units, $\Omega(k)$--- Berry curvature, B(x) slow magnetic field):

$$\dot{x} = {\partial E \over \partial k} + \dot{k}\times \Omega(k)$$

$$\dot{k} = - {\partial W\over \partial x} - \dot{x} \times B(x) $$

given by Sundaram and Niu, PRB 59 14915, 1999 for motion of electrons in bands with Berry curvature. The Lagrangian of these equations (when B=0) is the Lagrangian of this paper, as $\Omega$ is to $p$ as $B$ is to $x$. These semiclassical equations motivated Murakami to describe intrinsic spin Hall conductance (http://arxiv.org/abs/cond-mat/0405003).

The states of a periodic potential labelled by a wavenumber and a bandnumber, k and n, and the Berry phase tells you the phase you get when you adiabatically transition between neighboring k's. The phase for an adiabatic transition between neighboring k's is $A(k)\cdot dk$, and the phase for an infinitesimal loop at $k$ per unit area is the Berry curvature form $\Omega(k)$, just as the B form in electromagnetism, with k taking the place of x, and all transitions adiabatic.

From this, you can understand the Sundaram and Niu equations of motion very inuitively, much the same way as you can understand the quantum origin of the classical equations of motion in a magnetic field: a line of wavefronts with velocity v to the right in a constant magnetic field upward bend in the direction of changing wavefunction phase, since the lines of constant phase are systematically shifted due to the phase holonomy of the B field. The same hold for a line of k-space wavefronts with velocity $\dot{k}$ given by the k-space group velocity. In the presence of Berry curvature, the wavefronts bend in the direction of $\dot{k}\times \Omega$. So the Berry connection and curvature is indeed a 90 degree x-p phase space rotation from the usual situation of magnetic fields. So while the usual Bloch band Lagrangian with a substitution of $x-A(k)$ serves as the starting point for the study, and their results are accurate and derived correctly, this should not be considered a new result.

I will focus attention on the new results, and on the suggestions for future experiments. These concern the Hofstadter butterfly, a single band hopping model on a lattice with a constant magnetic field. This is considered in a trap which provides a slow quadratic potential in x space. This quadratic x-potential is a usual kinetic term after the 90 degree phase space rotation, and the Berry curvature provides, at small values of the Harper-Hofstadter magnetic field, a constant magnetic field on the band torus.

As they say, this precise model has been recently considered also in another publication (http://www.physik.uni-kl.de/agfleischhauer/dokuwiki/lib/exe/fetch.php?media=publications:physreva.89.033607.pdf ), but this reference did not consider the dual magnetic field picture, and in hindsight, after reading this paper, this is certainly the best way to analyze the model. Their theoretical insight from the dual formulation is substantial, they describe the details of the precise motion of the eigenvalues in response to changing magnetic field from the dual picture, they have a detailed description in terms of the Landau levels on the Brillouin torus. These add to the understanding of the model. All the results are accurate, as best as I can determine.

Their suggestion for a future publication regarding nonabelian Berry bands is interesting. Their comments about the experimental realizations look ok (although I don't know how exactly they plan to study non-toroidal manifolds). They gave the maximally insightful picture of the model, in a way amenable to quantum analog simulation by BEC. There are a lot of interesting things to do, I think.

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